The Dawn of Fluid Dynamics -...

The Dawn of Fluid DynamicsA Discipline between Science and Technology

Michael Eckert

WILEY-VCH Verlag GmbH & Co. KGaA

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Michael EckertThe Dawn of Fluid DynamicsA Discipline between Scienceand Technology


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The Dawn of Fluid DynamicsA Discipline between Science and Technology

Michael Eckert

WILEY-VCH Verlag GmbH & Co. KGaA


The author of this book

Dr. Michael EckertDeutsches Museum Mnchenemail: [email protected]

Cover illustration Wake downstream of a thin plate soaked in a water flow by Henri Werl, with kind permissionfrom ONERA, http://www.onera.fr

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V

Contents

Preface IX

1 Diverging Trends before the Twentieth Century 11.1 Galileos Abstraction 11.2 Hogs Bladders in St. Pauls Cathedral 61.3 Ballistics 91.4 DAlemberts Paradox 131.5 New Attempts to Account for Fluid Friction 151.6 Revival of Ideal Fluid Theory 181.7 Reynoldss Investigations of Direct or Sinuous Flow 221.8 Hydraulics and Aerodynamics: A Turn Towards Empiricism 241.9 Fluid Mechanics ca. 1900 28

2 The Beginnings of Fluid Dynamics in Gttingen, 19041914 312.1 Prandtls Route to Boundary Layer Theory 322.2 Per Experimentum et Inductionem Omnia 382.3 The First Doctoral Dissertations on Boundary Layers 422.4 Airship Research 452.5 The Discovery of the Turbulent Boundary Layer 492.6 The Beginnings of Airfoil Theory 52

3 Aviation and the Rise of Aerodynamics in the First World War 573.1 A Symbiotic Relationship 593.2 War Contracts 633.3 Gttingen Profiles 673.4 Max Munk and the Foundation of Airfoil Theory 733.5 Theory and Practice in Airplane Design 76

VI Contents

4 The Internationalization of Fluid Mechanics in the 1920s 834.1 American Emissaries at Prandtls Institute 844.2 Standardization 914.3 International Conferences 964.4 Applied Mathematics and Mechanics: A New International

Discipline 1004.5 Internationality in Practice: Max Munk at the NACA 102

5 A Working Program for Research on Turbulence 1075.1 Turbulent Pipe Flow 1085.2 Prandtls Research Program on Turbulence 1115.3 The Mixing Length Concept for the Fully Developed

Turbulence 1165.4 A Kind of Olympic Games 1185.5 Wind Tunnel Turbulence 124

6 Aerodynamics Comes of Age 1296.1 How Aerodynamics Became Institutionalized at Technical

Universities 1296.2 Glider Flight 1346.3 Krmn and Junkers: The Beginnings of Industrial Consulting in

Aeronautics 1386.4 Profile Measurements 1426.5 Airfoil Theory 148

7 New Applications 1537.1 Gas Dynamics 1547.2 Cavitation 1627.3 Meteorological and Geophysical Fluid Dynamics 1677.4 The Scope of Fluid Dynamics by the Early 1930s 174

8 Prandtl, Fluid Dynamics and National Socialism 1778.1 Preparing for War: Increased Funding for Prandtls Institute 1798.2 Aeronautical Science as an Instrument of Nazi Propaganda 1878.3 Goodwill Ambassador 190

9 New Centers 1959.1 Aachen 1969.2 Pasadena 2019.3 Zurich 207

Contents VII

10 Fluid Dynamics on the Eve of the Second World War 21310.1 Airfoil Theory 21310.2 Turbulence 22010.3 Gas Dynamics 228

11 Epilogue 233

Appendix

Abbreviations 243

References 245

Author Index 271

Name Index 275

Subject Index 279

IX

Preface

A leading representative of fluid dynamics defined this discipline as part ofapplied mathematics, of physics, of many branches of engineering, certainlycivil, mechanical, chemical, and aeronautical engineering, and of naval archi-tecture and geophysics, with astrophysics and biological and physiologicalfluid dynamics to be added. [1, p. 4]

Fluid mechanics has not always been as versatile as this definition suggests.Fifty years ago, astrophysical, biological, and physiological fluid dynamicswas still in the future. A hundred years ago, aeronautical engineering did notyet exist; when the first airplanes appeared in the sky before the First WorldWar, the science that became known as aerodynamics was still in its infancy.By the end of the 19th century, fluid mechanics meant hydrodynamics or hy-draulics: the former usually dealt with the aspects of ideal, i.e., frictionless,fluids, based on Eulers equations of motion; the latter was concerned withthe real flow of water in pipes and canals. Hydrodynamics belonged to thedomain of mathematics and theoretical physics; hydraulics, by contrast, wasa technology based on empirical rules rather than scientific principles. Theo-retical hydrodynamics and practical hydraulics pursued their own divergingcourses; there was only a minimal overlap, and when applied to specific prob-lems, the results could contradict one another [2].

This book is concerned with the history of fluid dynamics in the twentiethcentury before the Second World War. This was the era when fluid dynamicsevolved into a powerful engineering science. A future study will account forthe subsequent period, when this discipline acquired the multifaceted charac-ter to which the above quote alluded. The crucial era for bridging the prover-bial gap between theory and practice, however, was the earlier period, i.e.,the first four decades of the twentieth century. We may call these decades theage of Prandtl, because no other individual contributed more to the forma-tion of modern fluid dynamics. We may even pinpoint the year and the eventwith which this process began: it started in 1904, when Ludwig Prandtl pre-sented at a conference the boundary layer theory for fluids with little friction.Prandtls publication was regarded as one of the most extraordinary papers

X Preface

of this century, and probably of many centuries [1]it marked an epoch inthe history of fluid mechanics, opening the way for understanding the motionof real fluids [3].

In order to avoid any misunderstanding: this is not a biography of Prandtl,however desirable an account of Prandtls life might be. Nor is it a hero story;I do not claim that the emergence of modern fluid dynamics is due solely toPrandtl. If Prandtl and his Gttingen circles work is pursued here in moredetail than that of other key figures of this discipline, it is because the nar-rative needs a thread to link its parts, and Prandtls contributions provideenough coherence for this purpose. The history of fluid dynamics in the ageof Prandtl, as presented in the following account, is particularly a narrativeabout how science and technology interacted with another in the twentiethcentury. How does one account for such a complex process? In contrast tosociological approaches I pursue the history of fluid dynamics not within atheoretical model of sciencetechnology interactions. Nevertheless, the rela-tionship of theory and practice, science and engineering, or whatever rhetoricis used to refer to these antagonistic and yet so similar twins, implicitly runs asa recurrent theme through all chapters of this book. I share with philosophers,sociologists, and other analysts of science studies the concern to better graspsciencetechnology interactions, but I cannot see how to present the history offluid dynamics from the perspective of an abstract model. My own approachis descriptive rather than analytical; I approach the history of fluid dynamicsfrom the perspective of a narrator who is more interested in a rich portrayalof historical contexts than in gathering elements for an epistemological analy-sis. This approach requires deviations here and there from the main alley, soto speak, in order to clarify pertinent contexts, but I am conscious not to losethe narrative thread and regard as pertinent only what contributes to a betterunderstanding of the theorypractice issue. I postpone further reflections tothe epilogue, when this issue may be better discussed in view of the empiricalmaterial presented throughout the remainder of the book.

Many people and institutions have contributed to this work. Instead of ac-knowledging their help here individually in the form of a long list of names,I refer readers to the notes in the appendix, where readers may better appre-ciate how archives and authors of other studies helped to add flesh to theskeleton of my narrative. The only exceptions concern my colleagues fromthe Deutsches Museum and the Munich Center for the History of Science andTechnology, whom I owe thank for years of fruitful collaboration and stimu-lating discussions, and the Deutsche Forschungsgemeinschaft for funding theResearch Group 393, which formed the framework of this study.

Michael Eckert, Munich, May 2005

1

1Diverging Trends before the Twentieth Century

The flow of water or air around an obstacle is such a familiar phenomenon thatwe tend to underrate its importance in the history of science and technology.Throughout the centuries, the behavior of a body in a fluid was a fundamentaltheoretical problem and an obvious practical concern. The motion of celestialbodies, ships, projectiles, and other phenomena involved conceptions of fluiddynamics. Although the development of science from Aristotle to Einsteinis usually presented without excursions into the history of fluid dynamics,concepts about motion inevitably involve assumptions about fluid resistance.

1.1Galileos Abstraction

In Aristotles natural philosophy, the medium through which a motion pro-ceeded played a paradoxical role. In order to sustain the motion, a motiveagency was required. Aristotle (384322 BC)imagined that this motive agencyresided in the medium: We must, therefore, Aristotle wrote in Book VIII ofhis Physics, hold that the original movent gives the power of causing motionto air, or water, or anything else which is naturally adapted for being a moventas well as for being moved [4, p. 506]. At the start of the motion of a projec-tile, the medium would be displaced by the projectile, and together with thisdisplacement, a motive force would be passed along the trajectory. Thus, themedium acquired the power to propel the projectile. At the same time themedium would resist the motion: If air is twice as tenuous as water, Aristo-tle argued, the same moving body will spend twice as much time in travel-ling a certain path in water as in travelling the same path in air [5, p. 21].

Aristotle dominated pre-modern natural philosophy but some of hisviews also served as bones of contention. How could the same mediumat the same time propel and resist the motion of a projectile? Most famousamong those who criticized this concept was Jean Buridan (13001358), whoargued that the propulsive property resided in the projectile itself rather than

2 1 Diverging Trends before the Twentieth Century

in the medium. He called this property impetus: Whenever some agencysets a body in motion, Buridan wrote, it imparts to it a certain impetus, acertain power which is able to move the body along in the direction imposedupon it at the outset (...) It is this impetus which moves a stone after it hasbeen thrown until the motion is at an end. But because of the resistance of theair and also because of the heaviness, which inclines the motion of the stone ina direction different from that in which the impetus is effective, this impetuscontinually decreases [5, pp. 4950]. Now the medium through which themotion proceeded was left with just one property: resistance.

The impetus concept marked the emergence of the modern notions of in-ertia and momentum. But that did not happen at once. Even Galileo Galilei(15641642), with whom we associate the revolutionary turn from the me-dieval philosophy to the new science of motion, still mixed Aristotelian con-cepts with modern concepts of motion. Like his predecessors, Galileo strug-gled with the role of the medium through which a body moves. His famousDialogues Concerning Two New Sciences reveals what problems were behind theeffort to imagine how a body would move without the resistive property of themedium. Galileo lets Salviati ask, for example, What would happen if bod-ies of different weight were placed in media with different resistances? Theanswer was presented by comparing the motion in air and water: I found,Salviati continues, that the differences in speed were greater in those mediawhich were more resistant, that is, less yielding. This difference was such thattwo bodies which differed scarcely at all in their speed through air would, inwater, fall the one with a speed ten times as great as that of the other [6, p. 68].

Obviously, motion in air would be closer to a motion without any influ-ence of the medium. But there were problems in quantitatively measuringdifferences for bodies with different weights in air. It occurred to me there-fore, Galileo argues with the voice of Salviati, to repeat many times the fallthrough a small height in such a way that I might accumulate all those smallintervals of time that elapse between the arrival of the heavy and light bodiesrespectively at their common terminus. With the repetition of the free fall, hemeant the repeated swings of a pendulum:

Accordingly I took two balls, one of lead and one of cork, theformer more than a hundred times heavier than the latter, and sus-pended them by means of two equal fine threads, each four or fivecubits long. Pulling each ball aside from the perpendicular, I letthem go at the same instant, and they, falling along the circum-ferences of circles having these equal strings for semi-diameters,passed beyond the perpendicular and returned along the samepath. This free vibration repeated a hundred times showed clearlythat the heavy body maintains so nearly the period of the lightbody that neither in a hundred swings nor even in a thousand will

1.1 Galileos Abstraction 3

the former anticipate the latter by as much as a single moment,so perfectly do they keep step. We can also observe the effect ofthe medium which, by the resistance which it offers to motion, di-minishes the vibration of the cork more than that of the lead, butwithout altering the frequency of either; even when the arc tra-versed by the cork did not exceed five or six degrees while thatof the lead was fifty or sixty, the swings were performed in equaltimes [6, pp. 8485].

In order to find out how the resistance of air depends on the velocity, Galileocompared the swings of pendulums with equal weights but different ampli-tudes. He found that the air resistance is proportional to the velocity of themoving body [6, p. 254].

Already Galileos contemporaries noticed that these conclusions could nothave resulted from actual experiments. Marin Mersenne (15881648) com-pared the swings of equal pendulums with different amplitudes: he foundthat one which started swinging with an amplitude of two feet differed fromone with an amplitude of one foot already after thirty periods of oscillationby as much as one full period. In 1639, a year after the publication of theDialogues Concerning Two New Sciences, he remarked that if Galileo had per-formed real pendulum experiments and only waited for thirty or forty swings,he would have noticed the difference [7]. Recent pendulum experiments con-firmed Mersennes critique [8, 11].

This and other observations of Galileo stirred considerable debate amonghistorians of science to what extent did Galileo actually perform experi-ments? Only his pendulum experiments with small amplitude are presumedreal; those with larger amplitudes are regarded as imaginary or hypo-thetical, i.e., they were not performed in reality, but (contrary to mere thoughtexperiments) are based on extrapolation from empirical observations [7]. Ear-lier interpretations tended to categorize Galileos style of research into one oftwo extremes: either as deductive in the tradition of Platonic and idealisticnatural philosophy, in which the experiment only plays a role as a confirma-tion of insights gained by mere thinking; or as inductive, with the experimentas the origin of new knowledge. According to more recent historical studies,however, Galileos science was more complex and does not fit neatly into onecategory or the other alone [9].

The question whether Galileo actually performed free fall experiments fromthe leaning tower of Pisa attracted particular scrutiny [10]. As with the pen-dulum experiments, the problem of resistance plays an important role here,too. Aristotle says that an iron ball of one hundred pounds falling from aheight of one hundred cubits reaches the ground before a one-pound ball hasfallen a single cubit. I say, Salviati responds to such an obvious discrepancywith reality, that they arrive at the same time. You find, on making the exper-


iment, that the larger outstrips the smaller by two finger-breadths [6, p. 64].However, a modern calculation, which takes into account the air resistance,yields a difference of 1.05 m for the free fall of a 100-lb. iron sphere (with aradius of 11.13 cm) and a 1-lb. iron sphere (with a radius of 2.4 cm) over adistance of 100 cubits (58.4 m). The lighter sphere would be more than onemeter behind the heavier one certainly much more than the two finger-breadths in Galileos argument [11]. If Galileo really performed the towerexperiment, why didnt he notice this discrepancy? The puzzle can be re-solved by a psycho-physical argument: when an experimenter intends to re-lease simultaneously two different weights from his outstretched hands, thepalm with the lighter weight tends to open a bit earlier than the palm with theheavier weight; this difference could have compensated for the difference dueto the air resistance [12], [13, Supplement 3].

But Galileo, presumably, was rather motivated by a theoretical argument.The medium had to be thrust aside by the falling body, Salviati argued.This quiet, yielding, fluid medium opposes motion through it with a resis-tance which is proportional to the rapidity with which the medium must giveway to the passage of the body. By such reasoning, Galileo related the dis-placed mass of the medium to the resistance: And since it is known that theeffect of the medium is to diminish the weight of the body by the weight ofthe medium displaced, we may accomplish our purpose by diminishing injust this proportion the speeds of the falling bodies, which in a non-resistingmedium we have assumed to be equal [6, pp. 7475].

In other words, despite a flawed concept of fluid resistance in terms of buoy-ancy, Galileo arrived at his goal: the abstraction of a motion in a non-resistingmedium. With a vanishing buoyancy, the resistance would vanish too. Inthis case, with no mass to be displaced, all bodies would fall in the samemanner. Galileos law of free fall certainly has to be rated among the mostimportant accomplishments in the history of science, but it is erroneous to in-fer from Galileos abstraction that he had a correct notion of air resistance,as a widely read book on the history of aerodynamics has claimed [14, p. 8].Galileo did not aim at a theory of aerodynamics; his predominant concern wasAristotles natural philosophy. The abstraction of a motion in a non-resistingmedium, perceived as a motion in which no medium had to be displaced,touched upon another ancient philosophical belief: Aristotle believed in theimpossibility of a vacuum; for Galileo, it was the domain in which the laws offree fall hold. Maybe it is not an exaggeration to state that Galileos elabora-tions on the medium through which a body moves only served to justify hisabstraction of a motion in empty space.

Against this background it does not come as a surprise that it was a pupilof Galileo, Evangelista Torricelli (16081647), who is credited with presentingthe first experimental evidence of a vacuum. Torricelli emptied glass tubes

1.1 Galileos Abstraction 5

filled with mercury into a container, such that the openings of the tubes werenot exposed to the air. Inside the inverted tube, above a remaining column ofmercury, there was left an empty space, a Torricellian vacuum. The heightof the mercury column in the tube was found to depend on the ambient airpressure. Torricelli undertook these experiments with another pupil of Galileo(Vincenzio Viviani). Like Galileo himself, his pupils also were primarily inter-ested in refuting Aristotelian dogmas. Many have said [that vacuum] cannothappen, Torricelli wrote to another follower of Galileo after his experiment;yet, it may occur with no difficulty, and with no resistance from nature.Thus, he refuted the dogma of a horror vacui. He concluded, with a nowfamous quote: We live submerged at the bottom of an ocean of elementaryair which is known by incontestable experiments to have weight. [15, p. 84].

After Torricellis experiment the old debate among vacuists and plenistsseemed to be decided in favor of the vacuists, but Ren Descartes (15861650) renewed the belief of a universal filling of space. He denied thatGalileos extrapolation of free fall in empty space was based on sound ar-guments. According to Descartes doctrine, all natural phenomena resultedfrom the motions of infinitely fine weightless particles of an ether that per-vaded the entire universe. The particles of ordinary matter, such as air orwater, were supposed to have weight, so that their displacement by a movingbody would retard its motion. In order to prevent a temporary depletion be-hind a moving object, the displacement of matter involved a flow around theobject, which Descartes imagined as vortical. He extended his doctrine to theentire universe. The solar system was supposed to be an enormous vortex ofmatter, in which the planets orbited as smaller vortices around the center [16].

Descartes did not produce quantitative results neither for his cosmogonynor for the domain of earthly physics. Once, he communicated in a letter aformula about the retardation of a free falling body in a medium, whereby thespeed approached a limit in the form of an infinite geometrical series, but hedid not provide a physical argument for this result [23, p. 110]. Nevertheless,he exerted a remarkable influence on seventeenth century natural philosophy.Christiaan Huygens (16291695) pursued several of Descartes ideas, such asthe concept of an attracting force due to vortical motion around a center. Sucha force would keep a planet embedded in a vortex in his orbit around thesun. In order to illustrate this force, Huygens arranged a little sphere in acylindrical vessel filled with water such that it was free to move in a radialdirection only. When the vessel was rotated around its axis, the sphere movedinwards against the centrifugal force [16, pp. 7677].


1.2Hogs Bladders in St. Pauls Cathedral

Descartes concepts of motion also influenced Isaac Newton (16431727),but as an opponent rather than as a follower. Alluding to Descartes Prin-cipia Philosophiae, Newton titled his own three-volume treatise on mechanicsPhilosophiae Naturalis Principia Mathematica. The first volume with Newtonslaws of motion for a body in a vacuum is celebrated as the foundation ofclassical mechanics. It is less known that Newton also spent a lot of timedeveloping the laws of motion for a body in a fluid. The entire second volumeis dedicated to this problem. It was regarded as the most original part ofthe whole work, though also largely incorrect [17, p. 167]. As for Galileoand Descartes, the debate among vacuists and plenists was also a ma-jor issue for Newton. One of his pupils, Henry Pemberton, wrote in 1728 abook titled A View of Sir Isaac Newtons Philosophy about the second volumeof Newtons Principia: By this theory of the resistance of fluids, and theseexperiments our author decides the question so long agitated among naturalphilosophers whether the space is absolutely full of matter. The Aristoteliansand Cartesians both assert this plenitude; the Atomists have maintained thecontrary. Our author has chosen to determine this question by his theory ofresistance [18, p. 314].

If the universe were filled with a material substance, as taught by Descartesand his school, then the planets would encounter a resistance along their or-bits around the sun. Descartes vortex conception could not escape that funda-mental problem and therefore would have given rise to contradictions if it hadbeen formulated in a quantitative manner. Newton presented an alternativeconcept with his theory of universal gravitation, which assumed an emptyspace between the celestial bodiesor a bodiless medium that would not ex-ert a noticeable resistance: And therefore the celestial spaces, thro which theglobes of the Planets and Comets are perpetually passing towards all parts,with the utmost freedom, and without the least sensible diminution of theirmotion, must be utterly void of any corporeal fluid, excepting perhaps someextremely rare vapours, and the rays of light. This was Newtons conclusionat the end of the section Of the motion of fluids and the resistance made toprojected bodies [19, vol. 2, proposition 40, pp. 161162].

From the outset, Newton assumed: In mediums void of all tenacity, the re-sistances made to bodies are in the duplicate ratio of the velocities. Galileosrelation that the resistance is in the ratio of the velocity, according to New-ton, was more a mathematical hypothesis than a physical one [19, vol. 2,proposition 4, p. 11]. Others had already made the same assumption of aquadratic velocity dependence, which seemed to be more in agreement withempirical observations (see below); but Newton was the first natural philoso-pher who attempted to justify this relation on the basis of a physical model.

1.2 Hogs Bladders in St. Pauls Cathedral 7

His concept is too complex for a short summary. It may suffice to hint at New-tons argument for an elastic fluid like air, which he conceived as a gas ofparticles. Based on certain assumptions about the mutual collisions of theseparticles, Newton obtained quantitative results about the resistance of such afluid. But whether elastic fluids do really consist of particles so repelling eachother, he concluded, is a physical question. We have here demonstratedmathematically the property of fluids consisting of particles of this kind, thathence philosophers may take occasion to discuss that question [19, vol. 2,proposition 23, theorem 18, p. 79]. Newton explicitly envisioned differentsources of resistance, as from the expansion of the particles after the man-ner of wool, or the boughs of trees, or any other cause, by which the particlesare hindered from moving freely among themselves; the resistance, by rea-son of the lesser fluidity of the medium, will be greater than in the corollariesabove [19, vol. 2, proposition 34, theorem 17, p. 117]. He also developed anotion of viscosity: The resistance, arising from the want of lubricity in theparts of fluid, is, ceteris paribus, proportional to the velocity with which theparts of the fluid are separated from each other [19, vol. 2, proposition 51,p. 184].1

Newton did not content himself with establishing theorems. In order toinvestigate the resistances of fluids from experiments, I procured a squarewooden vessel (...) this I filled with rain-water: and having provided globesmade up of wax, and lead included therein, I noted the times of descents.Thus, Newton described the beginning of a series of experiments on fluid re-sistance. He used a pendulum with an oscillation period of a half-second forthe measurement of time, and meticulously compared the various outcomeswith his theoretical formulae: Three equal globes, weighing 141 grains in airand 4 3/8 in water, being let fall several times, fell in the times of 61, 62, 63,64 and 65 oscillations, describing a space of 182 inches, he described one ofthese experiments. And by the theory they ought to have fallen in 64 1/2oscillations, nearly. He noticed that sometimes the globes in falling oscil-late a little and believed that for this reason the resistance was somewhatgreater than in the duplicate ratio of the velocity. But in general he regardedthe outcome as an experimental verification of his square law formula for theresistance of a globe moving though a perfectly fluid compressed medium.After a series of 12 experiments he concluded that the theory agrees withthe phaenomena of bodies falling in water; it remains that we examine thephaenomena of bodies falling in air [19, vol. 2, proposition 40, pp. 145155].

1) In modern terms, this is equivalent to a linear relation between shearstress and strain rate: we call fluids with such viscous behaviorNewtonian. However, Newton did not investigate the relationbetween stress and strain. See [20, pp. 258259].


In order to verify his theory of the resistance of a spherical body movingin air, Newton, like Galileo, first performed pendulum experiments. He sus-pended a sphere by a fine thread on a hook, then varied the diameters andmaterials of the sphere, as well as the lengths of the thread. According tohis theory, the resistance was proportional to the square of the velocity of thesphere, and this is what he nearly observed. But he could not account forthe additional resistance of the thread which was certainly considerable. Healso compared the oscillations of the pendulum in air with those in water, butfound the outcome not reliable because the vessel in which the water was con-tained was not large enough so that by its narrowness [the vessel] obstructedthe motion of the water as it yielded to the oscillating globe. Even less con-clusive were pendulum experiments in mercury. I intended to have repeatedthese experiments with larger vessels, and in melted metals, and other liquorsboth cold and hot: but I had not leisure to try all, Newton admitted [19, vol. 2,proposition 31, pp. 95110].

Newton hoped to obtain more reliable measurements of air resistance withfree fall experiments: From the top of St. Pauls church in London in June1710 there were let fall together two glass globes, one full of quicksilver, theother of air. The two spheres traversed a height of 220 English feet (67 m) be-fore they shattered into pieces on the cathedrals floor. They were released bya sophisticated trapdoor-mechanism which ensured their simultaneous beginof fall. The time was measured by a pendulum with a period of oscillation ofone second. The experiment was repeated several times with varying weights.The spheres filled with mercury had a diameter of 0.8 inches; those filled withair were between 5.0 and 5.2 inches in diameter. The time of free fall was 4 sfor the heavier spheres and between 8 and 8.5 s for the lighter ones. (In a vac-uum the time would have been 3.7 s.) In order to compare these results withhis theory, Newton compared the experimental height of free fall with the dis-tance they would have traversed within the measured time according to hisformula. Both distances differed by less than 11 feet [19, vol. 2, proposition40, pp. 155157].

Newton mentioned in his Principia yet another series of free fall experimentsfrom a somewhat greater height in St. Pauls cathedral: Anno 1719 in themonth of July, Dr. Desaguliers made some experiments of this kind again, byforming hogs bladders into spherical orbs; which was done by means of aconcave wooden sphere, which the bladders, being wetted well fist, were putinto. After that, being blown full of air, they were obliged to fill up the spher-ical cavity that contained them; and then, when dry, were taken out. Thesewere let fall from the lantern on the top of the cupola of the same church;namely from a height of 272 feet. For comparison, a leaden sphere was letfall down at the same time. The air filled hogs bladders had diameters ofabout 5 inches and required about 20 s to fall down; the leaden spheres, by

1.3 Ballistics 9

contrast, reached the ground in 4 1/4 s. Newton also reported about phenom-ena which delayed the free fall by as much as a whole second sometimes, be-cause the bladders did not always fall directly down, but sometimes fluttereda little in the air, and waved to and fro as they were descending. One blad-der was wrinkled, and by its wrinkles was a little retarded. Nevertheless,he found that the results agreed much better with his theory than nine yearsago: Our theory therefore exhibits rightly, within a very little, all the resis-tance that globes moving either in air or in water meet with; which appears tobe proportional to the densities of the fluids in globes of equal velocities andmagnitudes [19, vol. 2, proposition 40, pp. 157159].

In modern terminology, Newtons formula for the resistance of a fluid isexpressed as D2v2, with representing the density of the fluid, D the di-ameter of the sphere, and v its velocity. This has become known as Newtonssquare law and has been established as a valid description of fluid resistancefor a wide range of flow regimes. However, although Newtons experimentsseemed to corroborate this law, they bear little evidence for Newtons theorybecause only one quantity, the time of free fall, was observed. The particlemodel gave rise to contradictory results when applied to bodies of differentshape in fluids such as air and water. In air, a rare medium, consisting ofequal particles freely disposed at equal distances from each other, the resis-tance of a sphere would be half of that of a cylinder with the same radiusmoving in the direction of it axis. In water, a compressed, infinite, and non-elastic fluid, would both experience the same resistance [19, vol. 2, proposi-tions 34 and 37, pp. 117, 135, and 141]. If Newton had compared experimentalresults of spheres and cylinders, he would have noticed a contradiction withhis theoretical results. Similarly, if he had calculated by the same reasoningthe air resistance of a flat plane oriented at an oblique angle to the flow ofair, he would have found a result proportional to the square of the sine of theangle of incidence. Newton did not perform such a calculation, but amongaerodynamicists Newtons sine square law became famous as an erroneousformula for the lift of a wing. At the beginning of the nineteenth century, thisformula was even used to demonstrate the impossibility of flying, and lateraerodynamicists blamed Newton for having delayed aviation at least for halfa century [18, p. 311].

1.3Ballistics

Beyond its pertinence to natural philosophy, the resistance of a body in amedium had always been a practical problem. Since antiquity, understandingthe trajectory of projectiles was an obvious challenge for natural philosophersas much as for practically minded men. It was part of a science named bal-


listics (derived from the Greek word , to throw). Niccol Tartaglia(14991557), a mathematician with some experience in military affairs, de-scribed the knowledge of his epoch on ballistics in a treatise Nova scientia.This work exerted some influence on Galileo, who spent considerable time inhis youth coming to grips with ballistics problems. In Tartaglias treatise, onecould read, for example, that a projectile traveled the farthest when fired atan angle of 45 degrees; but the trajectory was no parabola: on a horizontalplane, the distance between the vertex of the projectile and the site of its im-pact on the ground was always shorter than the distance between the originof its trajectory and the vertex, as depicted in Fig. 1.1. Initially, he adoptedthe Aristotelian belief that a trajectory starts out straight, but in a subsequentwork, he argued that the trajectories are curved everywhere [23, 24].

Fig. 1.1 Tartaglia imagined that a projectiles trajectory starts outstraight due to the violent motion impressed by the shot; it is followedby a curved mixed motion, and finally becomes natural [21, p. 38].

As recent studies of Galileos manuscripts have shown [25], Galileo de-scribed the trajectory of a projectile in a vacuum as a parabola before he ar-rived at his law of free fall not the other way around, as a deductive ap-proach would suggest. The parabola emerged in 1592, when Galileo lecturedon military technology at the University of Padua. Based on his convictionthat the air resistance did not exert an appreciable effect, Galileo assumed asymmetric trajectory in contrast to Tartaglias more realistic descriptions ofasymmetric trajectories. But when Torricelli derived ballistic tables based onparabolic trajectories in 1644, an artillery officer uttered doubts: he wrote toTorricelli that if it were not for the authority of the great Galileo, whom herevered, he would not believe that the motion of projectiles is parabolic. Torri-

1.3 Ballistics 11

celli admitted that if there are discrepancies, one should find out what causedthem, but with the experimental and theoretical tools available at the time ahundred years before the advent of calculus such efforts were futile. Thecorrespondence between the practical artillery officer and the theorist (Torri-celli was court mathematician at the Medicis) ended without a tangible resultshortly before Torricelli died in 1647 [26].

Fig. 1.2 Like the jets of a fountain [22, p. 325], ballistic trajectories wereassumed to be parabolic.

Throughout the seventeenth century it was fashionable for practical gun-ners to assume, like Galileo and Torricelli, that the parabola is the true tra-jectory of a projectile despite air resistance. Francois Blondel (16171686), afield marshal of the Royal French Army, published a treatise on LArt de Jetterles Bombes in which he addressed the problem of air resistance but assumedthat it could be neglected. He pointed to fountains with their nearly parabolicjets as evidence for this assumption see Fig. 1.2. Another treatise on TheGenuine Use and Effects of the Gunne, published in 1674, also claimed that airresistance is negligible and, therefore, a parabola describes the real trajectoryof a projectile. It is ironic that those who believed that air resistance does ex-ert a considerable influence and that the resulting trajectory is different froma parabola were not the gunners with experience with real trajectories, butmen like Huygens and Newton, who based their arguments on mathematicsrather than practical observations. Newton derived from the square law forair resistance that a parabolic trajectory would require that the air density notbe constant but become negative along part of its trajectory. A hyperbolic tra-jectory would not result in such a blatant contradiction. Therefore, It is evi-dent that the line which a projectile describes in a uniformly resisting medium,approaches nearer to these hyperbolas than to a parabola [23, pp. 120127,140141].


In retrospect, it is not astonishing why ballistic theory and practice divergedto such an extent before the eighteenth century. Only in 1742, with the treatiseby Benjamin Robins (17071751), New Principles of Gunnery, did a method be-come known by which it was possible to measure the velocity of a projectilein the beginning of its trajectory: the ballistic pendulum (see Fig. 1.3).

Fig. 1.3 Robinss ballistic pendulum.

Robinss innovative contributions to experimental ballistics made him fa-mous as Father Gunnery. He experimented with projectiles that left his gunwith velocities as high as 1,700 feet per second (559 m/s). If such a projectile,when fired at an angle of 45 degrees, would follow a parabolic trajectory, itwould hit the ground 17 miles away in contrast to an actual range of onlyabout half a mile. Projectiles with such a high starting velocity obviously ex-perienced an enormous resistance if their range was so much shorter. Thetrack described by the flight of shot or shells is neither a parabola, Robinsconcluded, nor nearly a parabola, unless they are projected with small veloc-

1.4 DAlemberts Paradox 13

ities. Robins also invented a whirling arm technique to measure the air resis-tance of objects with small velocities. Based on his experiments, he found thatall the theories of resistance hitherto established, are extremely defective [27,pp. 153154]. But Robinss mathematical abilities were limited. It was left toLeonhard Euler (17071783), in his German translation of Robinss New Prin-ciples of Gunnery, to elaborate a theory of ballistic trajectories [20, pp. 211220].

Besides Euler, mathematicians and natural philosophers like JohannBernoulli (16671748) and Jean le Rond dAlembert (17171783) becamedeeply engaged in ballistic calculations. The problem to find a projectilestrajectory became a proving ground for the newly developed calculus.2

1.4DAlemberts Paradox

The same eighteenth century thinkers who had recognized that air resistanceposed a serious problem for calculating the trajectory of a projectile alsoformulated the laws of motion for ideal, i.e., inviscid, fluids, and found astrange result. DAlembert published a treatise in 1768 titled Paradoxe pro-pos aux Gometres sur la Rsistance des Fluides in which he asserted thata body moving through an ideal fluid does not experience a resistive force.DAlembert, from his efforts in ballistics, knew about the practical importanceof air resistance, but neither he nor other theorists were able to derive New-tons square law or any other law of fluid resistance from the laws of me-chanics. In retrospect, dAlemberts paradox does not appear so paradoxicalbecause it was derived under the assumption of an inviscid fluid. Yet, it isdifficult to understand why the displacement of the fluid does not involve aforce. Euler had expressed this strange result many years before dAlembert,after whom the paradox finally became named. If each fluid particle flowedaround the body in such a way that it maintained the direction it had whenit was in front of the body, argued Euler in one of his comments to RobinssNew Principles of Gunnery, then there is no net force and the body would notexperience any resistance [20, p. 245].

DAlemberts paradox, therefore, should have entered the history of fluidmechanics more appropriately as the EulerdAlembert paradox. Both hadapproached these problems as theorists. Their mutual relation was often oneof fierce rivalry, which gave rise to some legendary stories. DAlemberts rep-

2) The major problem, however, was due to yet unknown physical pro-cesses rather than an unavailability of mathematical tools. At thehigh (usually supersonic) velocities of projectiles fired from can-nons and guns, the density of the air around the moving body is nolonger constant. The study of air resistance in varying air densityhad to await twentieth century gas dynamics.


utation has been overrated, claimed one historian of mechanics, while Eulersrole in this history was not appreciated enough.

Despite his theoretical leanings, Euler was very open-minded about prac-tical problems. Nevertheless, his practical work was regarded largely as afailure. When Frederick the Great, king of Prussia, gave orders to decoratehis Royal Garden of Sanssouci with water art, Euler became involved withhydraulic calculations about pumps and pipes required to raise water into anelevated water reservoir from where it was supposed to feed the fountains inthe park. The king, however, never came to enjoy a fountain. He blamed Eulerfor having failed miserably: My mill was constructed mathematically, and itcould not raise one drop of water to a distance of fifty feet from the basin.Vanity of Vanities! Vanity of mathematics. Based on this passage, historiansconcluded The mathematical genius Euler was a second-rate physicist, orEulers theory was not applicable for practical ends. This is how Euler isseen in the history of science as a prime example of the proverbial schismbetween theory and practice. However, although it is true that the water artconstructions in the Royal Garden of Sanssouci were abandoned unfinished inthe lifetime of Frederick the Great, this fact was not Eulers fault, but was theresult of the kings stinginess. He employed cheap laborers who had no expe-rience with such work and who completely ignored Eulers hydraulic advice,which could have prevented the sad outcome. Euler conceived a theory ofpipe flow that explained why the pipes always burst before water was raisedto the elevated reservoir: as a consequence of the pumping action, which ac-celerated the water through the pipes, the walls of the pipes had to sustain amuch higher water pressure than expected from the height difference betweenthe pumps and the reservoir [28].

This was not the first incidence that a study in fluid flow was motivatedby problems with water art. The science of moving water was among thosespecialties that were met with the greatest interest from Royal Academies.One outstanding work on hydraulics, which resulted from the patronage ofthe Paris Acadmie Royale des Sciences, is Edme Mariottes Trait de Mouve-ment des Eaux, published in 1686. Mariotte and other academy members per-formed experiments investigating the speed with which water is ejected froma pipe, the principles of raising water, the height of water jets, and the resis-tance of a body as a function of the flow velocity. The motivation to undertakesuch experiments came from ambitious projects of water constructions, suchas the canals all across France and the plans for the Royal Park at Versailles,where the worlds most sophisticated water art was established for the plea-sure of the Sun King and his court. The flow of water in open canals and inclosed pipes became the subject matter of intensive study. The law of energyconservation in fluids, Bernoullis equation, was formulated in the context ofpipe flow by Johann Bernoulli (16671748) and his son Daniel Bernoulli (1700

1.5 New Attempts to Account for Fluid Friction 15

1782). Like Euler, the Bernoullis are mainly renowned for their mathematicalwork, but as is evident from the fathers Hydraulica (1732) and the sons Hydro-dynamica (1738), their work was motivated to a large extent by practical con-cerns of contemporary water art. In the age of Euler and the Bernoullis, thenotions of hydrodynamics and hydraulics were used almost synonymously,often with an emphasis on the art of raising water and the several ma-chines employed for that purpose, as siphons, pumps, syringes, fountains,jets deau, fire-engines, etc., according to a contemporary dictionary [29].

After he had established and solved the equations of fluid motion for thespecial case of pipe flow, Euler formulated the general equations of motionfor inviscid fluids. They were published in 1755 under the title Principesgnraux du mouvement des fluides; with Eulers equations, as they werecalled, fluid mechanics was based on a firm theoretical foundation. Althoughthese equations are valid for ideal fluids only, which inevitably involvesdAlemberts paradox, a number of practical problems can still be solved onthat assumption.

1.5New Attempts to Account for Fluid Friction

In 1822, Claude Louis Marie Henri Navier (17851836) added a term to Eulersequations, which turned them into equations of motion for viscous fluids. Afew years later, Simon-Denis Poisson (17811840) arrived at the same result.Other contributors to this new formulation of the theory of fluid flow are Au-gustin Louis Cauchy (17891857) and Barr de Saint-Venant (17971886). Butonly in 1845 did George Gabriel Stokes (18191903) present a valid derivationfor the NavierStokes equations, as they became known. The earlier theo-ries of Navier and Poisson were based on hypotheses of atoms which, froma modern perspective, have to be dismissed as wrong, illustrating a com-mon phenomenon in the history of science: Falsehood Truth, commenteda twentieth century expert on fluid mechanics and historian of mechanics onthe gradual emergence of the NavierStokes equations, but then the treatiseof Stokes appeared as a burst of sunlight [30, p. 316].

It is not accidental that it was mostly scientists in post-revolutionary Francewho paid so much attention to the mechanics of continuous media not onlyfluid mechanics but also elasticity theory in the early nineteenth century.This interest was rooted in the Laplacian program, in which all phenomenain nature were believed to be explainable in terms of an attraction or repul-sion of particles. This program emerged in the tradition of Newtons naturalphilosophy: inspired by the model of celestial mechanics, central forces werebelieved to govern phenomena on a large scale as much as they do on the


scale of atoms [31]. However, there was little unanimity on how to pursuethis program: Navier was adhering the school of analytical mechanics, incontrast to another faction which headed for a more physical mechanics ap-proach. Institutionally these traditions were rooted in lcole Polytechniqueand the special engineering schools, lcole des Mines and lcole des Ponts etChausses. Navier, for example, had studied at lcole Polytechnique and atlcole des Ponts et Chausses before he became a professor himself at theseinstitutions. From a sociological perspective, his career was described as anearly example of a hybrid career, where the realms of science and technol-ogy became entangled [32].

Stokess effort to account for friction was also initially based on assump-tions about ultimate particles, but he became aware that his conclusionsdid not depend on such assumptions [33]. Like Navier, Stokes was primar-ily a theorist, but in contrast to Navier, he was not affiliated institutionallywith engineering. As a professor at the University of Cambridge, Stokes hadno official research interests devoted to experimental or technological studies.Nevertheless for Stokes mathematics was the servant and assistant, not themaster. His approach was described in an obituary: His guiding star in sci-ence was natural philosophy. Sound, light, radiant heat, chemistry, were hisfields of labour, which he cultivated by studying properties of matter with theaid of experimental and mathematical investigation [34].

For Stokes, like for other nineteenth century natural philosophers, hydrody-namics was a specialty where fundamental questions about the constitution ofmatter sometimes went hand in hand with practical problems. This dual ori-entation, which led to the NavierStokes equations, is also apparent in thederivation of what is known as Stokess law: a sphere of radius a moving witha constant velocity V in a fluid of viscosity experiences a resistance 6aV.Stokes arrived at this result by simplifying the NavierStokes equation so thatterms involving the square of the velocity were neglected. It was published in1850 in a paper titled On the Effect of the Internal Friction of Fluids on theMotion of Pendulums [35, vol. 3, 1141].

The relation to pendulums hints at the practical context that motivated thisstudy: from Galileo via Huygens and Newton until the nineteenth century,the pendulum was the preferred instrument to measure time, but the preci-sion that could be obtained theoretically was, in the true sense of the word,dampened by air resistance. When Stokes started to analyze the potential rea-sons why the swings of a pendulum would slow down, he investigated thebuoyancy that the sphere at the end of a pendulum experiences in a mediumas a primary cause. A second cause was the dynamic effect of the displace-ment of the medium, which resulted in an apparent increase of the inertia ofthe sphere. Stokes concluded that the mass which we must suppose addedto that of the pendulum is equal to half the mass of the fluid displaced. With

1.5 New Attempts to Account for Fluid Friction 17

regard to friction, it was unclear to what extent the density played a role.3

Experiments commissioned by the Board of Longitude had shown that the re-sistance depended both on the density and composition of the gas in whichthe pendulum swung. In practice, medium-related influences for a pendulumdesigned for a certain period of oscillation were accounted for in terms of cor-rection factors for the ideal length of a pendulum in vacuum. There werenumerous theoretical and experimental studies in order to determine suchcorrecting factors. Stokes cited studies performed by the German astronomerFriedrich Wilhelm Bessel (17841846) or the Frenchman Louis Gabriel Dubuat(17321787), whose research had been largely ignored by those interested inpendulum clocks, as Stokes argued, probably because such persons were notlikely to seek in a treatise on hydraulics for information connected with thesubject of their researches. Dubuat had, in fact, rather applied the pendulumto hydrodynamics than hydrodynamics to the pendulum. The same may besaid about Stokes. His goal was to derive an index of friction, by which theexperimentally determined correction factors for pendulums used for precisemeasurement of time could be understood in terms of hydrodynamics.

Stokess law was of interest far beyond its original pendulum context.Stokes argued, for example, that the resistance of the water droplets in a cloudmay be estimated from his law. The terminal velocity thus obtained is sosmall in the case of small globules such as those of which we may conceivea cloud to be composed, that the apparent suspension of the clouds does notseem to present any difficulty, he argued. The pendulum thus, in additionto its other uses, affords us some interesting information relating to the de-partment of meteorology [35, p. 10].

Stokes had also sketched another application which could be analyzed bythe NavierStokes equations: he derived a formula for the velocity profile ofa fluid in a tube. If one assumes that the velocity is zero at the inner wall ofthe tube (which Stokes mentioned as a possible assumption but did not pur-sue), one finds a parabolic increase of the velocity towards the tubes center.Integration over the tubes cross section yields the total flow as proportionalto r4 (with r being the radius of the tube), or the resistance per unit lengthas proportional to 1/r4. This law was found earlier from experiments by theGerman hydraulic engineer Gotthilf Hagen (17971884) and the French phys-iologist Jean Louis Poiseuille (17971869); it became known as the HagenPoiseuille law. Hagens experiments were performed with metal tubes witha diameter of a few centimeters and were motivated by practical considera-tions concerning the design of water pipelines. Poiseuille experimented with

3) Stokes assumed that the viscosity is proportional to , where is the index of friction, and is the density of the medium; it waslater shown by Maxwell that contrary to Stokess assumption, theviscosity is independent of the density and therefore, the densitydoes not enter into the formula of Stokess law.


glass tubes with a diameter of only a tenth of a millimeter; he aimed at a betterunderstanding of blood circulation [36, 37].

The theoretical explanation of the HagenPoiseuille law was published in1860 in a physiological as well as a physical context, the former in the Archivfr Anatomie, Physiologie und Medizin and the latter in the Annalen der Physik.What is remarkable about these publications is that the result stemmed fromsuch diverse disciplines physiology and physicswhich seems to indicatethat after the NavierStokes equations were formulated and the first applica-tions appeared, theory and practice would grow closer together. However,this was not the case. Hydrodynamics became an ever more theoretical sci-ence and hydraulics a specialty for practical men. The interest in the theoryof ideal fluids did not fade away but further increased when mathematiciansand physicists explored new avenues of fluid behavior in the second half ofthe nineteenth century.

1.6Revival of Ideal Fluid Theory

Despite dAlemberts paradox, there is an influence upon the motion of a bodyin an ideal fluid that is due to the displacement of the fluid. In 1852, themathematician Gustav Lejeune Dirichlet (18051859) investigated this influ-ence through a novel analysis of Eulers equations. He wondered whetherthere were specific motions in which a resistance in an ideal fluid was theoret-ically possible. Dirichlet analyzed the case of a sphere in a uniformly acceler-ated fluid. He found that the sphere experiences a constant force proportionalto the ratio of the densities of the fluid and the sphere, and to the accelerat-ing force. This resistance was independent of the momentary flow velocityand disappeared with a vanishing acceleration, so that for the case of uniformmotion, dAlemberts paradox was established. Dirichlets resistance hadnothing to do with friction but was a mere inertial effect due to the displace-ment of fluid by the solid body, as analyzed by Stokes in his pendulum motionexperiments. It was most conspicuous when expressed in terms of the kineticenergy: compared with motion in a vacuum, the kinetic energy of the spherein the fluid was as if the motion involved an increased mass of the sphere.That mass corresponded to the mass of the fluid which had to be displaced bythe sphere [38].

Although Dirichlets result was derived from ideal fluid theory, it was im-portant for the understanding of fluid resistance in real fluids because itshowed how to discern forces due to inertial effects from friction. One istempted to conclude that his result stemmed from efforts to learn more aboutthe differences between ideal and real fluids, but that was not Dirichlets mo-

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